We would like to also emphasize that we have dealt here with single-particle Hamiltonian models. The delicate length scale competition required for the topological characterization of quasicrystals can be easily perturbed. For example, disorder or dissipation produce a self-energy broadening that will be able to close small spectral gaps associated with long-ranged deviations from periodicity, i.e., they truncate long-ranged length scale. Alternatively, perturbing quasicrystals with strong potentials that close spectral gaps arising from the quasiperiodic length scales competition can generate real-space curvatures and observed chiral edge modes, which are not related to the higher-dimensional topology discussed here [1]}.
Similarly to disorder and dissipation, many-body interactions have also been shown to act as a cut-off on the length scale at which the incommensuration prevails [2]}, [3]}. At the same time, new results recently appeared, where strongly-correlated quasiperiodic spin chains exhibit topological excitation gaps [4]} or robust topological edge modes that are strongly shaped by the interaction [5]}. Furthermore, anomalous inverse compressibility is predicted to occur due to inter-band interactions [6]}.
<FIGURE>